Goals Rotational quantities as vectors. Math: Cross Product. Angular momentum

Transcription

1 Physcs 106 Week 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap 11.2 to 3 Rotatonal quanttes as vectors Cross product Torque expressed as a vector Angular momentum defned Angular momentum as a vector Newton s second law n vector form 1 Goals Rotatonal quanttes as vectors Math: Cross Product Angular momentum 1

2 So far: smple (planar) geometres Rotatonal quanttes Δθ, ω, α, τ, etc represented by postve or negatve numbers Rotaton axs was specfed smply as CCW or CW Problems were 2 dmensonal wth a perpendcular rotaton axs Now: 3D geometres rotaton represented n full vector form Angular dsplacement, angular velocty, angular acceleraton as vectors, havng drecton The angular dsplacement, speed, and acceleraton ( θ, ω, α ) are vectors wth drecton. The drectons are gven by the rght-hand rule: Fngers of rght hand curl along the angular drecton (See Fg.) Then, the drecton of thumb s the drecton of the angular quantty. 2

3 Example: trad of unt vectors showng rotaton n x- y plane +z ω = ωkˆ +x ω +y A dsk rotates at 3 rad/s n xy plane as shown above. What s angular velocty vector? Torque as a vector? Torque vector s defned from poston vector and force vector, usng cross product. 3

4 Math: Cross Product Cross Product (Vector Product) two vectors a thrd vector normal to the plane they defne measures the component of one vector normal to the other θ = smaller angle between the vectors c a b = ab sn( θ ) a b = b a c = a cross product of any parallel vectors = zero cross product s a maxmum for perpendcular vectors cross products of Cartesan unt vectors: î î = ĵ ĵ = kˆ kˆ = 0 kˆ = î ĵ = ĵ î ĵ = kˆ î = î kˆ î = ĵj kˆ = kˆ ĵj j k c b θ b a Drecton s defned by rght hand rule. More About the Cross Product Commutatve rule does not apply. A B B A, but A B= B A, The dstrbutve rule: A x ( B + C) = A x B + A x C If you are famlar wth calculus, the dervatve of a cross product obeys the chan rule, but preserves the order d ( ) = d of the terms: A + d B A B B A dt dt dt 4

5 Cross products usng components and unt vectors A B= ( AB ) ˆ+ ( ) ˆ+ ( ) ˆ y z AB z y AB z x AB x z j AB x y AB y x k A B= ˆ ˆ j kˆ A A A x y z B B B x y z Or, you can use dstrbutve b rule and + î î = ĵ ĵ = kˆ kˆ = 0 kˆ = î ĵ = ĵ î ĵ = kˆ î = î kˆ î = ĵ kˆ = kˆ ĵ j k Calculatng cross products usng unt vectors Fnd: A B Where: A = 2ˆ+ 3 ˆj; B = ˆ+ 2ˆj 5

6 Torque as a Cross Product τ = r F The torque s the cross product of a force vector wth the poston vector to ts pont of applcaton. τ = r F sn( θ) = r F = r F The torque vector s perpendcular to the plane formed by the poston vector and the force vector (e.g., magne drawng them tal-to-tal) Rght Hand Rule: curl fngers from r to F, thumb ponts along torque. Superposton: p τ = τ = r F net all all (vector sum) Can have multple forces appled at multple ponts. Drecton of τ net s angular acceleraton axs Fndng a cross product 5.1. A partcle located at the poston vector r = (î + ĵ) (n meters) has a force Fˆ = ( 2 î + 3ĵ) N actng on t. The torque n N.m about the orgn s? A) 1 kˆ B) 5 kˆ C) - 1 kˆ D) - 5 kˆ E) 2î + 3ĵ What f Fˆ = ( 3 î + 3ĵ)? 6

7 Net torque example: multple forces at multple ponts F F = 2 N ˆ appled at R = -2m ˆj = 4 N k ˆ appled at R = 3m ˆ Fnd the net torque about the orgn: j k Angular momentum concepts & defnton - Lnear momentum: p = mv - Angular (Rotatonal) momentum: L = moment of nerta x angular velocty = Iω nerta speed lnear momentum lnear m v p=mv rotatonal I ω L=Iω Angular momentum vector: L = Iω rgd body angular momentum 7

8 Angular momentum of a bowlng ball 6.1. A bowlng ball s rotatng as shown about ts mass center axs. Fnd t s angular momentum about that axs, n kg.m 2 /s A) 4 B) ½ C) 7 D) 2 E) ¼ ω = 4 rad/s M = 5 kg r = ½ m I = 2/5 MR 2 L = Iω Angular momentum of a pont partcle 2 L = Iω = mr ω = mv r = mvr sn( ϕ) = mvr = r p r P r r v v φ v = ω r r : moment arm Note: L = 0 f v s parallel to r (radally n or out) Angular momentum vector for a pont partcle L r p = m(r v) 8

9 Angular momentum of a pont partcle O r p r p T θ p L= rp = rpsn θ = r p T p If r = ( r, r,0) p= ( px, py,0) x y L = (0,0, rp rp) x y y x Angular momentum for car 5.2. A car of mass 1000 kg moves wth a speed of 50 m/s on a crcular track of radus 100 m. What s the magntude of ts angular momentum (n kg m 2 /s) relatve to the center of the race track (pont P )? A) A B) C) D) E) P B 5.3. What would the angular momentum about pont P be f the car leaves the track at A and ends up at pont B wth the same velocty? A) Same as above B) Dfferent from above C) Not Enough Informaton L = r p = p r = pr sn( θ) 9

10 net Net angular momentum L = L + L + L Example: calculatng angular momentum for partcles PP *: Two objects are movng as shown n the fgure. What s ther total angular momentum about pont O? m 2 m 1 10